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Boundedness of solutions of a non-local reaction–diffusion model for adhesion in cell aggregation and cancer invasion

Published online by Cambridge University Press:  01 February 2009

JONATHAN A. SHERRATT ,
STEPHEN A. GOURLEY ,
NICOLA J. ARMSTRONG  and
KEVIN J. PAINTER
JONATHAN A. SHERRATT
Affiliation:
Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK email: jas@ma.hw.ac.uk, nicola_j_armstrong@hotmail.com, painter@ma.hw.ac.uk
STEPHEN A. GOURLEY
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UKS.Gourley@surrey.ac.uk
NICOLA J. ARMSTRONG
Affiliation:
Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK email: jas@ma.hw.ac.uk, nicola_j_armstrong@hotmail.com, painter@ma.hw.ac.uk
KEVIN J. PAINTER
Affiliation:
Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK email: jas@ma.hw.ac.uk, nicola_j_armstrong@hotmail.com, painter@ma.hw.ac.uk
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Abstract

Adhesion of cells to one another and their environment is an important regulator of many biological processes but has proved difficult to incorporate into continuum mathematical models. This paper develops further the new modelling approach proposed by Armstrong et al. (A continuum approach to modelling cell–cell adhesion, J. Theor. Biol. 243: 98–113, 2006). The models studied in the present paper use an integro-partial differential equation for cell behaviour, in which the integral represents the sensing by cells of their local environment. This enables an effective representation of cell–cell adhesion, as well as random cell movement, and cell proliferation. The authors use this modelling approach to investigate the ability of cell–cell adhesion to generate spatial patterns during cell aggregation. The model is also extended to give a new representation of cancer growth, whose solutions reflect the balance between cell–cell and cell–matrix adhesion in regulating cancer invasion. The non-local term in these models means that there is no standard theory from which one can deduce the boundedness required for biological realism: specifically, solutions for cell density must lie between zero and a positive density corresponding to close cell packing. Here the authors derive a number of conditions, each of which is sufficient for the required boundedness, and they demonstrate numerically that cell density increases above the upper bound for some parameter sets not satisfying these conditions. Finally the authors outline what they regard as the main mathematical challenges for future work on boundedness in models of this type.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 20 , Issue 1 , February 2009 , pp. 123 - 144
Copyright
Copyright © Cambridge University Press 2008

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References

Ala-aho, R. & Kähäri, V.-M. (2005) Collagenases in cancer. Biochimie 87, 273–286.CrossRefGoogle ScholarPubMed
Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K. & Walter, P. (2008) The Molecular Biology of the Cell, 5th ed.. Garland Science, New York.Google Scholar
Alibaud, N., Azérad, P. & Isèbe, D. (submitted) A non-monotone non-local conservation law for dune morphodynamics.Google Scholar
Anderson, A. R. A. (2005) A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion. IMA J. Math. Appl. Med. Biol. 22, 163–186.CrossRefGoogle ScholarPubMed
Anderson, A. R. A., Chaplain, M. A. J. & Rejniak, K. A. (eds.) (2007) Single-Cell-Based Models in Biology and Medicine. Birkhäuser, Basel.CrossRefGoogle Scholar
Anderson, A. R. A., Weaver, A. M., Cummings, P. T. & Quaranta, V. (2006) Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127, 905–915.CrossRefGoogle ScholarPubMed
Araujo, R. & McElwain, L. (2004) A history of the study of solid tumour growth: The contribution of mathematical modelling. Bull. Math. Biol. 66, 1039–1091.CrossRefGoogle Scholar
Armstrong, N. J., Painter, K. J. & Sherratt, J. A. (2006) A continuum approach to modelling cell–cell adhesion. J. Theor. Biol. 243, 98–113.CrossRefGoogle ScholarPubMed
Armstrong, N. J., Painter, K. J. & Sherratt, J. A. (in press) Adding adhesion to the cell cycle model for somite formation. Bull. Math. Biol.Google Scholar
Bauer, A. L., Jackson, T. L. & Jiang, Y. (2007) A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis. Biophys. J. 92, 3105–3121.CrossRefGoogle ScholarPubMed
Berrier, A. L. & Yamada, K. M. (2007) Cell–matrix adhesion. J. Cell. Physiol. 213, 565–573.CrossRefGoogle ScholarPubMed
Byrne, H. M. (1997) The importance of intercellular adhesion in the development of carcinomas. IMA J. Math. Appl. Med. Biol. 14, 305–323.CrossRefGoogle ScholarPubMed
Byrne, H. M. & Chaplain, M. A. J. (1996) Modelling the role of cell–cell adhesion in the growth and development of carcinomas. Math. Comp. Modelling 24 (12), 1–17.CrossRefGoogle Scholar
Chaplain, M. A. J. (1996) Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modelling of the stages of tumour development. Math. Comp. Modelling 23 (6), 47–87.CrossRefGoogle Scholar
Chaplain, M. A. J. & Lolas, G. (2005) Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system. Math. Models Methods Appl. Sci. 15, 1685–1734.CrossRefGoogle Scholar
Cheng, X., Den, Z. N. & Koch, P. J. (2005) Desmosomal cell adhesion in mammalian development. Eur. J. Cell Biol. 84, 215–223.CrossRefGoogle ScholarPubMed
Cristini, V., Lowengrub, J. & Nie, Q. (2003) Nonlinear simulation of tumour growth. J. Math. Biol. 46, 191–224.CrossRefGoogle Scholar
Dallon, J. C. & Othmer, H. G. (2004) How cellular movement determines the collective force generated by the Dictyostelium discoideum slug. J. Theor. Biol. 231, 203–222.CrossRefGoogle ScholarPubMed
Deutsch, A. & Dormann, S. (2005) Cellular Automaton Modeling of Biological Pattern Formation: Characterization, Applications, and Analysis. Birkhäuser, Boston.Google Scholar
Drasdo, D. & Forgacs, G. (2000) Modeling the interplay of generic and genetic mechanisms in cleavage, blastulation, and gastrulation. Dev. Dyn. 219, 182–191.3.3.CO;2-1>CrossRefGoogle ScholarPubMed
Drasdo, D. & Hohme, S. (2003) Individual-based approaches to birth and death in avascular tumors. Math. Comp. Modelling 37 (11), 1163–1175.CrossRefGoogle Scholar
Drasdo, D. & Hohme, S. (2005) A single-cell-based model of tumor growth in vitro: Monolayers and spheroids. Phys. Biol. 2, 133–147.CrossRefGoogle ScholarPubMed
Drasdo, D., Kree, R. & McCaskill, J. S. (1995) Monte-Carlo approach to tissue–cell populations. Phys. Rev. E 52, 6635–6657.CrossRefGoogle ScholarPubMed
Drasdo, D. & Loeffler, M. (2001) Individual-based models to growth and folding in one-layered tissues: Intestinal crypts and early development. Nonlinear Anal. 47, 245–256.CrossRefGoogle Scholar
Fogelson, A. L. (2007) Cell-based models of blood clotting. In: Anderson, A. R. A., Chaplain, M. A. J. & Rejniak, K. A. (editors), Single-Cell-Based Models in Biology and Medicine, Birkhäuser, Basel, pp. 243–269.CrossRefGoogle Scholar
Fogelson, A. L. & Guy, R. D. (2008) Immersed-boundary-type models of intravascular platelet aggregation. Comput. Methods Appl. Mech. Engng. 197, 2087–2104.CrossRefGoogle Scholar
Foty, R. A. & Steinberg, M. S. (2004) Cadherin-mediated cell–cell adhesion and tissue segregation in relation to malignancy. Int. J. Dev. Biol. 48, 397–409.CrossRefGoogle ScholarPubMed
Foty, R. A. & Steinberg, M. S. (2005) The differential adhesion hypothesis: A direct evaluation Dev. Biol. 278, 255–263.CrossRefGoogle ScholarPubMed
Frieboes, H. B., Lowengrub, J. S., Wise, S., Zheng, X., Macklin, P., Elaine, L. B. D. & Cristini, V. (2007) Computer simulation of glioma growth and morphology. Neuroimage 37 (Suppl. 1), S59–S70.CrossRefGoogle ScholarPubMed
Frieboes, H. B., Zheng, X., Sun, C.-H. & Tromberg, B. (2006) An integrated computational/experimental model of tumour invasion. Cancer Res. 66, 1597–1604.CrossRefGoogle Scholar
Friedman, A. (2007) Mathematical analysis and challenges arising from models of tumor growth. Math. Models Methods Appl. Sci. 17, 1751–1772.CrossRefGoogle Scholar
Galle, J., Aust, G., Schaller, G., Beyer, T. & Drasdo, D. (2006) Individual cell-based models of the spatial-temporal organization of multicellular systems – Achievements and limitations. Cytometry 69A, 704–710.CrossRefGoogle Scholar
Galle, J., Loeffler, M. & Drasdo, D. (2005) Modelling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro. Biophys. J. 88, 62–75.CrossRefGoogle ScholarPubMed
Gassmann, P., Enns, A. & Haier, J. (2004) Role of tumor cell adhesion and migration in organ-specific metastasis formation. Onkologie 27, 577–582.Google ScholarPubMed
Gerisch, A. (submitted) On the approximation and efficient evaluation of integral terms in PDE models of cell adhesion.Google Scholar
Gerisch, A. & Chaplain, M. A. J. (2008) Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion. J. Theor. Biol. 250, 684–704.CrossRefGoogle ScholarPubMed
Glazier, J. A. & Graner, F. (1993) Simulation of the differential adhesion driven rearrangement of biological cells. Phys. Rev. E 47, 2128–2154.CrossRefGoogle ScholarPubMed
Glazier, J. A., Raphael, R. C., Graner, F. & Sawada, Y. (1995) The energetics of cell sorting in three dimensions. In: Beysens, D., Forgacs, G. & Gaill, F. (editors), Interplay of Genetic and Physical Processes in the Development of Biological Form, World Scientific Publishing Company, Singapore, pp. 54–61.Google Scholar
Graner, F. & Glazier, J. A. (1992) Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys. Rev. Lett. 69, 2013–2016.CrossRefGoogle ScholarPubMed
Green, J. E. F., Waters, S. L., Shakesheff, K. M., Edelstein-Keshet, L. & Byrne, H. M. (in preparation) Non-local models for the interactions of hepatocytes and stellate cells during aggregation.Google Scholar
Grygierzec, W., Deutsch, A., Philipsen, L., Friedenberger, M. & Schubert, W. (2004) Modelling tumour cell population dynamics based on molecular adhesion assumptions. J. Biol. Syst. 12, 273–288.CrossRefGoogle Scholar
Halbleib, J. M. & Nelson, W. J. (2006) Cadherins in development: Cell adhesion, sorting, and tissue morphogenesis. Genes Dev. 20, 3199–3214.CrossRefGoogle ScholarPubMed
Hart, I. (2005) The spread of tumours. In: Knowles, M. & Selby, P. (editors), Introduction to the Cellular and Molecular Biology of Cancer, Oxford University Press, Oxford, UK, pp. 278–288.CrossRefGoogle Scholar
Hillen, T. & Painter, K. J. (2008) A users guide to PDE models for chemotaxis. J. Math. Biol. 58, 183–217.CrossRefGoogle ScholarPubMed
Jorgensen, P. & Tyers, M. (2004) How cells coordinate growth and division. Curr. Biol. 14, R1014–R1027.CrossRefGoogle ScholarPubMed
Kim, Y., Stolarska, M. & Othmer, H. G. (2007) A hybrid model for tumor spheroid growth in vitro I: Theoretical development and early results. Math. Models Methods Appl. Sci. 17, 1773–1798.CrossRefGoogle Scholar
Lock, J. G., Wehrle-Haller, B. & Stromblad, S. (2008) Cell–matrix adhesion complexes: Master control machinery of cell migration. Seminars Cancer Biol. 18, 65–76.CrossRefGoogle ScholarPubMed
Macklin, P. & Lowengrub, J. (2007) Nonlinear simulation of the effect of microenvironment on tumor growth. J. Theor. Biol. 245, 677–704. See also the erratum on p. 581 of vol. 247.CrossRefGoogle ScholarPubMed
Maree, A. F. & Hogeweg, P. (2002) Modelling Dictyostelium discoideum morphogenesis: The culmination. Bull. Math. Biol. 64, 327–353.CrossRefGoogle ScholarPubMed
Merks, R. M. H. & Glazier, J. A. (2005) A cell-centered approach to developmental biology. Physica A 352, 113–130.CrossRefGoogle Scholar
Mombach, J. C. M., Glazier, J. A., Raphael, R. C. & Zajac, M. (1995) Quantitative comparison between differential adhesion models and cell sorting in the presence and absence of fluctuations. Phys. Rev. Lett. 75, 2244–2247.CrossRefGoogle ScholarPubMed
Moreira, J. & Deutsch, A. (2005) Pigment pattern formation in zebrafish during late larval stages: A model based on local interactions. Dev. Dyn. 232, 33–42.CrossRefGoogle Scholar
Palsson, E. (2007) A 3-D deformable ellipsoidal cell model with cell adhesion and signalling. In: Anderson, A. R. A., Chaplain, M. A. J. & Rejniak, K. A. (editors), Single-Cell-Based Models in Biology and Medicine, Birkhäuser, Basel, pp. 271–299.CrossRefGoogle Scholar
Palsson, E. & Othmer, H. G. (2000) A model for individual and collective cell movement in Dictyostelium discoideum. Proc. Natl. Acad. Sci. USA 97, 10448–10453.CrossRefGoogle Scholar
Perumpanani, A. J., Sherratt, J. A., Norbury, J. & Byrne, H. M. (1996) Biological inferences from a mathematical model for malignant invasion. Invasion Metastasis 16, 209–221.Google ScholarPubMed
Reddig, P. J. & Juliano, R. L. (2005) Clinging to life: Cell to matrix adhesion and cell survival. Cancer Metastasis Rev. 24 425–439.CrossRefGoogle ScholarPubMed
Savill, N. J. & Hogeweg, P. (1997) Modelling morphogenesis: From single cells to crawling slugs. J. Theor. Biol. 184, 229–235.CrossRefGoogle ScholarPubMed
Savill, N. J. & Sherratt, J. A. (2003) Control of epidermal stem cell clusters by Notch-mediated lateral induction. Dev. Biol. 258, 141–153.CrossRefGoogle ScholarPubMed
Schaller, G. & Meyer-Hermann, M. (2007) A modelling approach towards epidermal homoeostasis control. J. Theor. Biol. 247, 554–573.CrossRefGoogle ScholarPubMed
Steinberg, M. S. (1962) On the mechanism of tissue reconstruction by dissociated cells, III. Free energy relations and the reorganization of fused, heteronomic tissue fragments. Proc. Natl. Acad. Sci. USA 48, 1769–1776.CrossRefGoogle ScholarPubMed
Steinberg, M. S. (2007) Differential adhesion in morphogenesis: A modern view. Curr. Op. Genetics Dev. 17, 281–286.CrossRefGoogle ScholarPubMed
Stott, E. L., Britton, N. F., Glazier, J. A. & Zajac, M. (1999) Stochastic simulation of benign avascular tumour growth using the Potts model. Math. Comp. Modelling 30 (5–6), 183–198.CrossRefGoogle Scholar
Turner, S. & Sherratt, J. A. (2002) Intercellular adhesion and cancer invasion: A discrete simulation using the extended Potts model. J. Theor. Biol. 216, 85–100.CrossRefGoogle ScholarPubMed
Turner, S., Sherratt, J. A. & Cameron, D. (2004) Tamoxifen treatment failure in cancer and the nonlinear dynamics of TGFβ. J. Theor. Biol. 229, 101–111.CrossRefGoogle ScholarPubMed
Weiner, R., Schmitt, B. & Podhaisky, H. (1997) Rowmap – A row-code with Krylov techniques for large stiff odes. Appl. Num. Math. 25, 303–319.CrossRefGoogle Scholar
Zajac, M., Jones, G. L. & Glazier, J. A. (2000) Model of convergent extension in animal morphogenesis. Phys. Rev. Lett. 85, 2022–2025.CrossRefGoogle ScholarPubMed
Zigrino, P., Löffek, S. & Mauch, C. (2005) Tumor-stroma interactions: Their role in the control of tumor cell invasion. Biochimie 87, 321–328.CrossRefGoogle ScholarPubMed